nLab
oplax monoidal functor

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Definition

If CC and DD are monoidal categories, an oplax monoidal functor F:CDF : C \to D is defined to be a lax monoidal functor F:C opD opF: C^{op} \to D^{op}. So, among other things, tensor products are preserved up to morphisms of the following sort in DD:

Δ c,c:F(cc)F(c)F(c)\Delta_{c,c'} : F(c \otimes c') \to F(c) \otimes F(c')

which must satisfy a certain coherence law.

Properties

An oplax monoidal functor sends comonoids in CC to comonoids in DD, just as a lax monoidal functor sends monoids in CC to monoids in DD. For this reason an oplax monoidal functor is sometimes called a lax comonoidal functor. The other obvious terms, colax monoidal and lax opmonoidal, also exist (or at least are attested on Wikipedia).

Note that a strong opmonoidal functor –in which the morphisms ϕ\phi are required to be isomorphisms— is the same thing as a strong monoidal functor.

Proposition

A functor with a right adjoint is oplax monoidal if and only if that right adjoint is a lax monoidal functor.

Proof

This is a special case of the statement of doctrinal adjunction for the case of the 2-monad whose algebras are monoidal categories,

Here is the explicit construction of the oplax monoidal structure from a lax monoidal structure on a right adjoint:

Let (LR):CRLD(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D be a pair of adjoint functors and let (C,)(C,\otimes) and (D,)(D,\otimes) be structures of monoidal categories.

Then if RR is a lax monoidal functor LL becomes an oplax monoidal functor with oplax unit

L(I D)I C L(I_D) \to I_C

the adjunct of the lax unit I DR(I D)I_D \to R(I_D) of RR and with oplax monoidal transformation

(L(xy)Δ x,yL(x)L(y)) (L (x \otimes y) \stackrel{\Delta_{x,y}}{\to} L(x) \otimes L(y))

given by the adjunct of

xyη xη yRLxRLy Lx,LyR(LxLy). x \otimes y \stackrel{\eta_x \otimes \eta_y}{\to} R L x \otimes R L y \stackrel{\nabla_{L x, L y}}{\to} R(L x \otimes L y) \,.

By the formula for adjuncts in terms of the adjunction counit (this prop.) this adjunct is the composite

L(xy)L(η xη y)L(RLxRLy)L( Lx,Ly)LR(LxLy)ϵ LxLyLxLy. L(x \otimes y) \stackrel{L(\eta_x \otimes \eta_y)}{\longrightarrow} L(R L x \otimes R L y) \stackrel{L(\nabla_{L x, L y})}{\longrightarrow} L R(L x \otimes L y) \stackrel{\epsilon_{L x \otimes L y}}{\longrightarrow} L x \otimes L y \,.

This appears for instance on p. 17 of (SchwedeShipley).

References

The construction of oplax monoidal functors from right adjoint lax monoidal functors is considered for instance around page 17 of

Last revised on July 31, 2019 at 05:31:21. See the history of this page for a list of all contributions to it.